Find the volume of the solid bounded by the surface z =5 +(x-4)
^2+2y and the planes x = 3, y = 3 and coordinate planes.
a. First find the volume by actual calculation.
b. Estimate the volume by dividing the region into nine equal
squares and evaluating the functional value at the mid-point of the
respective squares and multiplying with the area and summing it.
Find the error from step a.
c. Then estimate the volume by dividing each...
1. For z=x^2-xy^2 find:
a. the gradient for z;
b. the directional derivative in direction of <3,-1>;
c.the approximation for Δz when taking a Δs =0.05 step from
(1,2) in direction of <3,-1>;
d. the approximation to the maximum Δz possible when taking a Δs
=0.05 step from (1,2).
Set up an integral that uses the disk method to find the volume
of the solid of revolution obtained by revolving the area between
the curves y = sech(x/2), y =2, x =0 and x = 4 around the line y=2.
Include a sketch of the region and show all work to integrate and.
Note: Recall that sech(u) = 1/cosh(u).
Please show details for every single step